Pseudoholomorphic tori in the Kodaira-Thurston manifold

TL;DR

This paper computes genus one Gromov-Witten invariants for pseudoholomorphic tori in the non-Kähler Kodaira-Thurston manifold, using a family of symplectic forms to overcome trivial invariants.

Contribution

It provides the first genus one Gromov-Witten computation for a non-Kähler manifold by analyzing a family of symplectic forms on the Kodaira-Thurston manifold.

Findings

Computed family Gromov-Witten invariants for pseudoholomorphic tori.

Established a method for non-Kähler manifolds using symplectic form families.

First example of genus one family Gromov-Witten invariants in this setting.

Abstract

The Kodaira-Thurston manifold is a quotient of a nilpotent Lie group by a cocompact lattice. We compute the family Gromov-Witten invariants which count pseudoholomorphic tori in the Kodaira-Thurston manifold. For a fixed symplectic form the Gromov-Witten invariant is trivial so we consider the twistor family of left-invariant symplectic forms which are orthogonal for some fixed metric on the Lie algebra. This family defines a loop in the space of symplectic forms. This is the first example of a genus one family Gromov-Witten computation for a non-K\"ahler manifold.